Divide sweet potatoes among four greased 13-in. x 9-in. baking dishes.
In a large saucepan over medium heat, combine the butter, sugar, milk, cinnamon and nutmeg; cook and stir until butter is melted and mixture is smooth. Remove from the heat; stir in coconut and vanilla. Pour over sweet potatoes.
Cover and bake at 350° for 45 minutes. Uncover; sprinkle each with 1-1/2 cups marshmallows. Bake 5-10 minutes longer or until marshmallows just begin to puff and melt.